Exact asymptotics of Laplace integrals on nonsmooth functions.
The author determines the exact asymptotics (including estimates for the remainder term) of the Laplace type integral \[ I(\lambda)=\int_X g(x) \exp(\lambda S(x))\,\mu(dx) \quad \text{ as } \lambda \to \infty\,. \] Here \((X,B;\mu)\) is a finite measure space, and \(g\) and \(S\) are real valued measurable functions such that \[ \text{ess sup}_{x\in X} S(x) < \infty, \quad \text{ess sup}_{x\in X} | g(x)| < \infty, \quad \mu\{x\in X: g(x)=0\}=0\,. \] In particular, no assumptions on the topological structure of \(X\) are made, and no smoothness conditions on \(g\) and \(S\) are posed. Some concrete examples and an application to probability theory are given.
scientific article; zbMATH DE number 2120998
scientific article